Answer
\[\begin{gathered}
{\mathbf{r}}''\left( t \right) = \left\langle { - \frac{1}{4}{{\left( {t + 4} \right)}^{ - 3/2}}, - 2{{\left( {t + 1} \right)}^{ - 3}},2{e^{ - {t^2}}}\left( {1 - 2{t^2}} \right)} \right\rangle \hfill \\
{\mathbf{r}}'''\left( t \right) = \left\langle {\frac{3}{8}{{\left( {t + 4} \right)}^{ - 5/2}},6{{\left( {t + 1} \right)}^{ - 4}}, - 4t{e^{ - {t^2}}}\left( {3 - 2{t^2}} \right)} \right\rangle \hfill \\
\end{gathered} \]
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = \sqrt {t + 4} {\bf{i}} + \frac{t}{{t + 1}}{\bf{j}} - {e^{ - {t^2}}}{\bf{k}} \cr
& {\text{Differentiate}} \cr
& {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {\sqrt {t + 4} } \right]{\bf{i}} + \frac{d}{{dt}}\left[ {\frac{t}{{t + 1}}} \right]{\bf{j}} - \frac{d}{{dt}}\left[ {{e^{ - {t^2}}}} \right]{\bf{k}} \cr
& {\bf{r}}'\left( t \right) = \frac{1}{{2\sqrt {t + 4} }}{\bf{i}} + \left( {\frac{{t + 1 - t}}{{{{\left( {t + 1} \right)}^2}}}} \right){\bf{j}} - \left( { - 2t{e^{ - {t^2}}}} \right){\bf{k}} \cr
& {\bf{r}}'\left( t \right) = \frac{1}{{2\sqrt {t + 4} }}{\bf{i}} + \frac{1}{{{{\left( {t + 1} \right)}^2}}}{\bf{j}} + 2t{e^{ - {t^2}}}{\bf{k}} \cr
& \cr
& {\bf{r}}''\left( t \right) = \frac{d}{{dt}}\left[ {\frac{1}{{2\sqrt {t + 4} }}} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {\frac{1}{{{{\left( {t + 1} \right)}^2}}}} \right]{\bf{j}} + \frac{d}{{dt}}\left[ {2t{e^{ - {t^2}}}} \right]{\bf{k}} \cr
& {\bf{r}}''\left( t \right) = - \frac{1}{4}{\left( {t + 4} \right)^{ - 3/2}}{\bf{i}} - 2{\left( {t + 1} \right)^{ - 3}}{\bf{j}} + \left( {2{e^{ - {t^2}}} - 4{t^2}{e^{ - {t^2}}}} \right){\bf{k}} \cr
& {\bf{r}}''\left( t \right) = \left\langle { - \frac{1}{4}{{\left( {t + 4} \right)}^{ - 3/2}}, - 2{{\left( {t + 1} \right)}^{ - 3}},2{e^{ - {t^2}}}\left( {1 - 2{t^2}} \right)} \right\rangle \cr
& \cr
& {\bf{r}}'''\left( t \right) = \frac{d}{{dt}}\left[ { - \frac{1}{4}{{\left( {t + 4} \right)}^{ - 3/2}}} \right]{\bf{i}} + \frac{d}{{dt}}\left[ { - 2{{\left( {t + 1} \right)}^{ - 3}}} \right]{\bf{j}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{d}{{dt}}\left[ {2{e^{ - {t^2}}}\left( {1 - 2{t^2}} \right)} \right]{\bf{k}} \cr
& {\bf{r}}'''\left( t \right) = \frac{3}{8}{\left( {t + 4} \right)^{ - 5/2}}{\bf{i}} + 6{\left( {t + 1} \right)^{ - 4}}{\bf{j}}\, + \left( { - 12t{e^{ - {t^2}}} + 8{t^{3{e^{ - {t^2}}}}}} \right){\bf{k}} \cr
& {\bf{r}}'''\left( t \right) = \left\langle {\frac{3}{8}{{\left( {t + 4} \right)}^{ - 5/2}},6{{\left( {t + 1} \right)}^{ - 4}}, - 4t{e^{ - {t^2}}}\left( {3 - 2{t^2}} \right)} \right\rangle \cr} $$
